Configurations of points and the symplectic Berry-Robbins problem
We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group Sp(n), instead of the Lie group U(n). Denote by h a Cartan algebra of Sp(n), and Δ the union of the zero sets of the roots of Sp(n) tensore...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146322 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Configurations of points and the symplectic Berry-Robbins problem / J. Malkoun // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862541740425084928 |
|---|---|
| author | Malkoun, J. |
| author_facet | Malkoun, J. |
| citation_txt | Configurations of points and the symplectic Berry-Robbins problem / J. Malkoun // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group Sp(n), instead of the Lie group U(n). Denote by h a Cartan algebra of Sp(n), and Δ the union of the zero sets of the roots of Sp(n) tensored with R3, each being a map from h⊗R3→R3. We wish to construct a map (h⊗R3)∖Δ→Sp(n)/Tn which is equivariant under the action of the Weyl group Wn of Sp(n) (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of Sp(n), and Tn is the diagonal n-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for n=2.
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| first_indexed | 2025-11-24T18:45:30Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146322 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T18:45:30Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Malkoun, J. 2019-02-08T20:54:30Z 2019-02-08T20:54:30Z 2014 Configurations of points and the symplectic Berry-Robbins problem / J. Malkoun // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 5 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 51F99; 17B22 https://nasplib.isofts.kiev.ua/handle/123456789/146322 We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group Sp(n), instead of the Lie group U(n). Denote by h a Cartan algebra of Sp(n), and Δ the union of the zero sets of the roots of Sp(n) tensored with R3, each being a map from h⊗R3→R3. We wish to construct a map (h⊗R3)∖Δ→Sp(n)/Tn which is equivariant under the action of the Weyl group Wn of Sp(n) (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of Sp(n), and Tn is the diagonal n-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for n=2. The author would like to thank Sir Michael Atiyah for kindly replying to his emails, and would
 like to thank the anonymous referees for all their suggestions, which ended up making the article
 much more readable. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Configurations of points and the symplectic Berry-Robbins problem Article published earlier |
| spellingShingle | Configurations of points and the symplectic Berry-Robbins problem Malkoun, J. |
| title | Configurations of points and the symplectic Berry-Robbins problem |
| title_full | Configurations of points and the symplectic Berry-Robbins problem |
| title_fullStr | Configurations of points and the symplectic Berry-Robbins problem |
| title_full_unstemmed | Configurations of points and the symplectic Berry-Robbins problem |
| title_short | Configurations of points and the symplectic Berry-Robbins problem |
| title_sort | configurations of points and the symplectic berry-robbins problem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146322 |
| work_keys_str_mv | AT malkounj configurationsofpointsandthesymplecticberryrobbinsproblem |